Question

Let \(\vec{a}\), \(\vec{b}\) and \(\vec{c}\) be three vectors such that \(|\vec{a}| = \sqrt{3}\), \(|\vec{b}| = 5\), \(\vec{b} \cdot \vec{c} = 10\) and the angle between \(\vec{b}\) and \(\vec{c}\) is \(\frac{\pi}{3}\). If \(\vec{a}\) is perpendicular to the vector \(\vec{b} \times \vec{c}\), then \(|\vec{a} \times (\vec{b} \times \vec{c})|\) is equal to ______.

Hint:

Find \(|\vec{c}|\) from \(\vec{b} \cdot \vec{c}\), then \(|\vec{b} \times \vec{c}|\); since \(\vec{a}\) is perpendicular to \(\vec{b} \times \vec{c}\), the magnitude is a plain product.

Answer 1

Correct Answer: 30

\(\vec{b} \cdot \vec{c} = |\vec{b}||\vec{c}|\cos\frac{\pi}{3} \Rightarrow 10 = 5|\vec{c}| \cdot \frac{1}{2} \Rightarrow |\vec{c}| = 4\).
\(|\vec{b} \times \vec{c}| = |\vec{b}||\vec{c}|\sin\frac{\pi}{3} = 5 \cdot 4 \cdot \frac{\sqrt{3}}{2} = 10\sqrt{3}\).
Since \(\vec{a} \perp (\vec{b} \times \vec{c})\), the angle between them is \(\frac{\pi}{2}\), so \(|\vec{a} \times (\vec{b} \times \vec{c})| = |\vec{a}||\vec{b} \times \vec{c}| = \sqrt{3} \cdot 10\sqrt{3} = 30\).
Answer: 30